If `x` and `y` satisfy the equation `y = 2[x]+9` and `y = 3[x+2]` simultaneously, the `[x+y]` is (where `[x]` is the greatest integer function)

Options

A. `21`
B. `18`
C. `30`
D. `12`

Explanation

To solve the problem, we need to find the values of \( x \) and \( y \) that satisfy the equations \( y = 2[x] + 9 \) and \( y = 3[x + 2] \) simultaneously, where \([x]\) is the greatest integer function (also known as the floor function).

### Step-by-Step Solution:

1. **Set the equations equal to each other:**
Since both expressions equal \( y \), we can set them equal to each other:
\[ 2[x] + 9 = 3([x] + 2) \]

2. **Expand the right-hand side:**
\[ 2[x] + 9 = 3[x] + 6 \]

3. **Rearrange the equation:**
Move all terms involving \([x]\) to one side and constant terms to the other:
\[ 2[x] - 3[x] = 6 - 9 \]
\[ -[x] = -3 \]

4. **Solve for \([x]\):**
Multiply both sides by -1:
\[ [x] = 3 \]

5. **Find \( y \):**
Substitute \([x] = 3\) back into either equation to find \( y \). Let's use the first equation:
\[ y = 2[3] + 9 = 2 \times 3 + 9 = 6 + 9 = 15 \]

6. **Calculate \([x + y]\):**
Now we need to find \([x + y]\):
\[ x + y = [x] + y = 3 + 15 = 18 \]

7. **Final answer:**
Therefore, \([x + y] = 18\).